Design of sliding mode power system stabilizer via ... - Semantic Scholar
Proceedings ZOO3 IEEE International Symposium on ComputationalIntelligence in Robotics and Automation July 16-20,2003,Kobe, Japan
Design of Sliding Mode Power System Stabilizer via Genetic Algorithm Tsong-Liang Huang Chih-Han Chang Ju-Lin Lee Hui-Mei Wang Department of Electrical Engineering Tamkang University #151, Ying-Chuan Rd. Tamsui, Taipei Hsien, Taiwan 25137, R.0.C Tel : +886-2-26215656 Ext 2615 Fax : +886-2-26221565 E-mail : [email protected] to change. In this paper, we use the two-level sliding mode power system dynamic stabilizer via genetic algorithm to stabilize the system. We use the genetic algorithm to lind the switching control signals and use sliding mode control to find the control signal of the generator. The advantages of the proposed method are illustrated by a numerical simulation of the multi-machine power systems. It appears that the proposed method reduces the oscillation and enhances the dynamic stability of the power system. Then, the proposed method will he compared with optimal control method and optimal reduced order method [4][5][6].
Abstract This paper proposes a new approach for combining genetic algorithm and sliding mode control to design the power system stabilizers (PSS). The design of a PSS can be formulated as an optimal linear regulator control problem. However, implementing this technique requires the design of estimators. This increases the implementation and reduces the reliability of control system. These reasons, therefore, favor a control scheme that uses only some desired state variables, such as torque angle and speed. To deal with this problem, we use the optimal reduced models to reduce the power system model into two state variables system by each generator. We use the genetic algorithm to find the switching control signals and use sliding mode control to find control signal of the generator. The advantages of the proposed method are illustrated by numerical simulation of the multi-machine power systems.
2 Two-Level Stabilization 2.1 Full Order Model
An example of the time optimal position control of a multi-machine system is used to illustrate the implementation and to evaluate the performance of the “OPEM algorithm.
Keywords : PSS, Sliding Mode, Genetic Algorithm
1 Introduction The power system stabilizers[l] are added to the power system to enhance the damping of the electric power system. The design of PSSs can be formulated as an optimal linear regulator control problem whose solution is a complete state control scheme [2]. But, the implementation requires the design of state estimators. These are the r e a ” that a control scheme uses only some desired state variables such torque angle and speed. Upon this, a scheme referred to as optimal reduced order model whose state variables are the deviation of torque angles and speeds will be used. The approach retains the modes that mostly affect these variables. The model is used to design an output states feedback controller. By using only the output feedback, the control strategy can be implemented easily.
Fig1 The multi-machine system The multi-machine power system full order model given by [6] is shown Fig. I.
where
x
[+
s,
v4
4
s2 ~ e ; ?AV,.*!
A : denotes deviation from operation point uj : denotes speed 6, : denotes torque angle
The traditional PSSs strategies adopt the previous information of the system to decide the control signal so that it is hard to control the power system before it is going
0-7803-7866-0/03/$17.00 02003 IEEE
(1)
i=Ax+Bu
e: : denotes voltage proportional to direct axis flux
linkages
515
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
V,
Consider the variable v is
: denotes generation field voltage
v = x2 + Q;:Q:,~,
2.2 lbo-level Optimal Design The two-level optimal design given in [4] is shown in Fig. 2.
Such that
Ki U!
Assume the equation of x, is as follows:
+ A12v
XI = Al*lx,
(7)
where AI', = A,, - A12Q:QL.
From eqn.(7), we can use the Riccati equation to fmd the constant positive d e f ~ t matrix e P.
PA,*,+ A;lP - PAl2Q;Q;P
Fig.2 The two-level optimal design
+ el*, =0
(8)
It is indicated from the simulation results that the system response is highly oscillatory. From eqn.(5) and (8)
3 Sliding Mode Control 3.1 Optimal Switching Hyperplane
When the sliding mode occurs, the state trajectories of the controlled system will be kept on the pre-specified switching hyperplane. The optimal switch hyperplane s is written as
Consider the linear time-invariant multivariable system described by the equation (2)
X=Ax+Bu
s = C'x = [ALP + Q;
where A E R""" B E R"""
- Q,,E
(11)
3.2 Switching Control Signal
In order to get the optimal switching hyperplane of the sliding mode controller, we defme the quadratic cost functionof J asfollows:
The input of the system is obtained as:
The control law can be considered separately by the two control terms and represented by Where Q=["" Q2i
Q'2],Q$ = e l 2 Q>O and f, is Q22
J=
~1, ( x ~ Q l +?xTQI2x2 lxl +x:QZx2 d)
(13)
u=Ueq+uh
the starting time which the sliding mode occurs.
where
(4)
U,,= - K : x
and uq =-K:x
is a discontinuous control signal,
is a linear equivalent control signal.
The feedback gain K , is appropriate chosen as
516
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
their desirable qualities through a random process. In this paper, the uniform crossover method is adopted. The procedure is to select a pair of strings from the mating pool at random, then, a mark is selected at random. Finally, two new strings are generated by swapping all characters correspond to the position of the mark where the hit is “1”. Although the crossover is done by random selection, it is not the same as a random search through the search space. Since it is based on the reproduction process, it is an effective means of exchanging information and combining portions of high-fitness solutions.
(14)
p, f 0 r s . q < 0 It is shown that if the gain parameters a, and p, are chosen so that s . S S 0 is satisfied, then the hitting will occur. When the state is on the sliding surface S = 0 , the purpose of the equivalent control is to keep the state staying on the sliding surface so it can be derived from setting the time derivative of s ,and S , equal to zero, that is (15)
U* = u’li.o
The feedback gain K ,
4.3 Mutation
is obtained as
K L = (C‘B -)C’.A 4 The Genetic Algorithms
GAS are searching techniques using the mechanics of natural selection and natural genetics for efficient global searches [9]. In comparison to the conventional searching algorithms, GAShas the following characteristics : (a) GAS work directly with the discrete points coded by finite length shings (chromosomes), not the real parameters themselves; (b) GAS consider a group of points (called a population size) in the search space in every iteration, not a single point; (c) GAS use fitness function information instead of derivatives or other auxiliary knowledge; and (d) GASuse probabilistic transition rules instead of deterministic rules. Generally, a simple GA consists of the three basic genetic operators : (a) Reproduction; (b) Crossover; and (c) Mutation. They are described as follows[lO]. 4.1 Reproduction
Mutation is a process to provide an occasional random alteration of the value at a particular string position. In the case of binary string, this simply means changing the state of a bit from 1 to 0 and vice versa In this paper we provide a uniform mutation method. This method is fust to produce a mask and select a string randomly, then complement the selected string value correspond to the position of mask where the bit value is “1”. Mutation is needed because some digits at particular position in all strings may be eliminated during the reproduction and the crossover operations. So the mutation plays the role of a safeguard in GAS. It can help GAS to avoid the possibility of mistaking a local optimum for a global optimum. 5 Results
The model given in [6] is X=Ax+Bu
where ~ 0 . 2 4 4 -0.0747 377 0
1:
Reproduction is a process to decide how many copies of individual strings should be produced in the mating pool according to their fitness value. The reproduction operation allows strings with higher fitness value to have larger number of copies, and the strings with lower fitness values have a relatively smaller number of copies or even none at all. This is an artificial version of natural selection (strings with higher fitness values will have more chances to survive).
B=
-0.1431
0
0
0.0747
0.Wl
0
0
0
0
0
0
0.046
0.13
398.58
-3967
-0.178
-0.146
0
-0.M
-0.455
0.244
0
-398.56
-19498.8
-50
0 0
0
0.178
-0.0433
0
-0.2473
0 0
0
0
0
376.99
0
0
0 0 0 0 0
0.056
0
0
-0.0565
-0.3061
0.149
0
-677.39
0
0
677.78
-133M.16
-50
ro o o
10
o o
0.1234 -10234.22
2500
o
o o 0 o o o
0
1’
25001
The eigenvalues of the power system are as follows:
-25.1741+j67.8187 - 0.0904 f j9.843 - 0.2443
4.2 Crossover
Crossover is a recombined operator for two high-fitness strings (parents) to produce two offsprings by matching
-25.23925 j30.3072 - 0.0006
To improve the system damping using the two-level scheme, the decomposed system and control matrices are
517
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
as follows: For system 1
Sliding
X , = [Am,
Aeql AVJ
-0.244 A , = [ 377 0 0 El =[0
Controller
-0,747 0 -0.046 -398.56
0 0 2500
-0.1431 0 -0.455 -19498.8
Reduced Order Model
I
A8.1
I L
0.244 -50.0
1'
Sliding Controller
Reduced Order Model
Using the expression given by [7], a reduced order model of system 1 is obtained. XI, = Fix,,
+ G,u,
(18)
where xlr = [Am, A8,
'1
GI =[-0.1826
1'
-0.734
4
=[
-0.26 377
-0.07
1,
Fig. 3 The structure of the proposed design
and
I n t h paper, we propose the genetic algorithm to fmd the switching surface vector and the switching control signals. Furthermore, in order to find the parameters to facilitate the controlled system with small integml absolute error, we define the following fitness function:
For system 2
x2=bo2 as, A,
=I
f =g1*g2
-0.2473 -0.177 377 0 0 -0.0565 0 677.78
4 =[O 0
AV~~J
kq2
2
0
-0.3061 -13364.1
0.1492
, i = 1,2, and
-50.0
0 2500'1
6=1
Using the expression given by [7], a reduced order model of system 2 is obtained.
The following parameters of GAS are considered
i2r = F2x2,+ Gzu2
1' -2.6202 1'
where xgr =[Am2 A6,
G, =[-0.2688
F2
=[
-0.18 377
-0.18
1,
Population size40 Crossover probability=0.9 Mutation probability=0.03 Chromosome length=30 Generations=2000 Range=[-250 2501
and
The global control matrix G is given by G=[
(20)
-0.146
0.118 -0.36 0.0975 -0.2974
0.02 -0.209 0.0141 -0.1465
From eqn.(ll), and eqn.(l6), we can get the optimal switching hyperplane C and the feedback gain K , of the power system.
1
For system 1
And the proposed design is shown in Fig. 3.
s, = [-3.0056 K:l =[-350.9
-1.6189b,, = C:x,, 0.12112
]
518
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
For system 2 s2 =[-1.457
Ks:2
=
-1.1052xl, =C:x,,
[- 126.66
0.079773
]
The transient responses of the angular frequencies with global control to a 5% change in the mechanical torque of system 1 and system 2 are shown in Fig.4 and Fig.5 respectively.
“.I
\ - - - --I
Fig3 (a) The torque angle response of system 1
Fig.4 (a) The angular frequency response of system 1
Fig.5 (b) The torque angle response of system 2 Fig.5 The transient responses of the torque angles 6 Summary
This paper proposes a new approach for combining genetic algorithm and sliding mode control to design the power system stabilizers. We use the genetic algorithm to fmd the switching surface vector and the switching control signals, and use the sliding mode control to fmd the control signal of the generator. By using the output feedback only, this approach reduces the implementation cost and the reliability of the power system. Comparison of the proposed method with the traditional optimal control method, the effectiveness of the proposed method in enhancing the dynamic performance stability is verified through the simulation results.
i
Fig.4 (b) The angular frequency response of system 2 Fig.4 The transient responses of the angular frequencies
References [l] Arcidiacono V, Ferrari E, and Saccomanno F.. Studies on damping electromechanical oscillation in multimachine system with longitudinal shucture, IEEE Transactions, PAS-95:450460, 1976. [2] Bauer D. L., Cory D.B., Buhr W. D., Ostroski G B., Cogswell S. S., and Swanson D. A.. Simulation of
519
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
low frequency undamped oscillations in large power systems, IEEE Transactions, PAS-89: 1239-1247, 1996. [3] Anderson P. M. and Fouad A. A.. Power system conhol and stabilig. Iowa State University Press, Ames, Iowa, 1977. [4] Y.L. Abdel-Magid, and GM. My. Two-level Stabilization in Multimachine Power Systems. Electric Power Systems Research, Vo1.6:33-41, 1983. [5] Feliachi A., Zbang X., and Sims C. S.. Power system stabilizers Design using Optimal Reduced Order Models Part I: Model Reduction. IEEE Transactions, PWRS-3: 1670-1675, 1988. [6] Feliachi A., Zhang X., and Sims C. S.. Power system stabilizers Design using Optimal Reduced Order Models Part LI: design. IEEE Transactions, PWRS-3: 1676-1684, 1988. 171 T.L. Huang, T.Y. Huang, and W.T. Yang. Two-level optimal output feedback stabilizer design. Power Systems, IEEE Transactions on , 6(3):1042-1048 , Aug. 1991. [SI Moussa H. A. and Yu Y. N., Optimal power system stabilization through excitation andlor govemor control. IEEE Transactions, PAS-91:1166-1173, 1972. [9] Do Bomfun, A.L.B., Taranto, GN., and Falcao, D.M.. Simultaneous tuning of power system damping controllers using genetic algorithms. Power Systems, IEEE Transactiom on , 15(1):163-169, Feb. 2000. [lOIShih-Yu Chang, Chih-Han Cbang, Chan-Sheng Lin, and Tsong-Liang Huang. Sliding Mode Control Design Using Fuzzy Genetic Algorithm. In 2000 Conference on Industrial Automatic Control and Power Application, pages El-17-El-22, Taiwan, December, 20m. [lI]Ren-Kuang Lung, Senjyi Chen, Wen-Shiow Kao, and Chen-Hao Lin. Study of Power System Stabilizer by Combining Genetic Algorithm and F u u y Control. In 2001 Ninth National Conference on Fuzzy Theory and Its Applications, pages 499-503, Taiwan, November 2001. [lZIChih-Han Chang, Shih-Yu Cbang, Ru-Lin Lee, Tsong-Liang Huang, and Ying-Tung Hsiao. Two-level Power System Dynamic Stabilizer Design via Sliding Mode Control. In 2002 Conference on Industrial Automatic Control and Power Application, pages C3-1C3-6, Taiwan, December, 2002. [13]Chh-Han Chang, KingTan Lee, Pei-Yi Lin, and Tsong-Liang Huang. Power System Dynamic Stabilizer Design via Combining Genetic Algorithm and Sliding Mode Control. In 2002 Tenth National Conference on Fuuy Theory and Its Applications, pages D4-31- D4-37, December, Taiwan, 2002.
520
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
Design of Sliding Mode Power System Stabilizer via Genetic Algorithm Tsong-Liang Huang Chih-Han Chang Ju-Lin Lee Hui-Mei Wang Department of Electrical Engineering Tamkang University #151, Ying-Chuan Rd. Tamsui, Taipei Hsien, Taiwan 25137, R.0.C Tel : +886-2-26215656 Ext 2615 Fax : +886-2-26221565 E-mail : [email protected] to change. In this paper, we use the two-level sliding mode power system dynamic stabilizer via genetic algorithm to stabilize the system. We use the genetic algorithm to lind the switching control signals and use sliding mode control to find the control signal of the generator. The advantages of the proposed method are illustrated by a numerical simulation of the multi-machine power systems. It appears that the proposed method reduces the oscillation and enhances the dynamic stability of the power system. Then, the proposed method will he compared with optimal control method and optimal reduced order method [4][5][6].
Abstract This paper proposes a new approach for combining genetic algorithm and sliding mode control to design the power system stabilizers (PSS). The design of a PSS can be formulated as an optimal linear regulator control problem. However, implementing this technique requires the design of estimators. This increases the implementation and reduces the reliability of control system. These reasons, therefore, favor a control scheme that uses only some desired state variables, such as torque angle and speed. To deal with this problem, we use the optimal reduced models to reduce the power system model into two state variables system by each generator. We use the genetic algorithm to find the switching control signals and use sliding mode control to find control signal of the generator. The advantages of the proposed method are illustrated by numerical simulation of the multi-machine power systems.
2 Two-Level Stabilization 2.1 Full Order Model
An example of the time optimal position control of a multi-machine system is used to illustrate the implementation and to evaluate the performance of the “OPEM algorithm.
Keywords : PSS, Sliding Mode, Genetic Algorithm
1 Introduction The power system stabilizers[l] are added to the power system to enhance the damping of the electric power system. The design of PSSs can be formulated as an optimal linear regulator control problem whose solution is a complete state control scheme [2]. But, the implementation requires the design of state estimators. These are the r e a ” that a control scheme uses only some desired state variables such torque angle and speed. Upon this, a scheme referred to as optimal reduced order model whose state variables are the deviation of torque angles and speeds will be used. The approach retains the modes that mostly affect these variables. The model is used to design an output states feedback controller. By using only the output feedback, the control strategy can be implemented easily.
Fig1 The multi-machine system The multi-machine power system full order model given by [6] is shown Fig. I.
where
x
[+
s,
v4
4
s2 ~ e ; ?AV,.*!
A : denotes deviation from operation point uj : denotes speed 6, : denotes torque angle
The traditional PSSs strategies adopt the previous information of the system to decide the control signal so that it is hard to control the power system before it is going
0-7803-7866-0/03/$17.00 02003 IEEE
(1)
i=Ax+Bu
e: : denotes voltage proportional to direct axis flux
linkages
515
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
V,
Consider the variable v is
: denotes generation field voltage
v = x2 + Q;:Q:,~,
2.2 lbo-level Optimal Design The two-level optimal design given in [4] is shown in Fig. 2.
Such that
Ki U!
Assume the equation of x, is as follows:
+ A12v
XI = Al*lx,
(7)
where AI', = A,, - A12Q:QL.
From eqn.(7), we can use the Riccati equation to fmd the constant positive d e f ~ t matrix e P.
PA,*,+ A;lP - PAl2Q;Q;P
Fig.2 The two-level optimal design
+ el*, =0
(8)
It is indicated from the simulation results that the system response is highly oscillatory. From eqn.(5) and (8)
3 Sliding Mode Control 3.1 Optimal Switching Hyperplane
When the sliding mode occurs, the state trajectories of the controlled system will be kept on the pre-specified switching hyperplane. The optimal switch hyperplane s is written as
Consider the linear time-invariant multivariable system described by the equation (2)
X=Ax+Bu
s = C'x = [ALP + Q;
where A E R""" B E R"""
- Q,,E
(11)
3.2 Switching Control Signal
In order to get the optimal switching hyperplane of the sliding mode controller, we defme the quadratic cost functionof J asfollows:
The input of the system is obtained as:
The control law can be considered separately by the two control terms and represented by Where Q=["" Q2i
Q'2],Q$ = e l 2 Q>O and f, is Q22
J=
~1, ( x ~ Q l +?xTQI2x2 lxl +x:QZx2 d)
(13)
u=Ueq+uh
the starting time which the sliding mode occurs.
where
(4)
U,,= - K : x
and uq =-K:x
is a discontinuous control signal,
is a linear equivalent control signal.
The feedback gain K , is appropriate chosen as
516
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
their desirable qualities through a random process. In this paper, the uniform crossover method is adopted. The procedure is to select a pair of strings from the mating pool at random, then, a mark is selected at random. Finally, two new strings are generated by swapping all characters correspond to the position of the mark where the hit is “1”. Although the crossover is done by random selection, it is not the same as a random search through the search space. Since it is based on the reproduction process, it is an effective means of exchanging information and combining portions of high-fitness solutions.
(14)
p, f 0 r s . q < 0 It is shown that if the gain parameters a, and p, are chosen so that s . S S 0 is satisfied, then the hitting will occur. When the state is on the sliding surface S = 0 , the purpose of the equivalent control is to keep the state staying on the sliding surface so it can be derived from setting the time derivative of s ,and S , equal to zero, that is (15)
U* = u’li.o
The feedback gain K ,
4.3 Mutation
is obtained as
K L = (C‘B -)C’.A 4 The Genetic Algorithms
GAS are searching techniques using the mechanics of natural selection and natural genetics for efficient global searches [9]. In comparison to the conventional searching algorithms, GAShas the following characteristics : (a) GAS work directly with the discrete points coded by finite length shings (chromosomes), not the real parameters themselves; (b) GAS consider a group of points (called a population size) in the search space in every iteration, not a single point; (c) GAS use fitness function information instead of derivatives or other auxiliary knowledge; and (d) GASuse probabilistic transition rules instead of deterministic rules. Generally, a simple GA consists of the three basic genetic operators : (a) Reproduction; (b) Crossover; and (c) Mutation. They are described as follows[lO]. 4.1 Reproduction
Mutation is a process to provide an occasional random alteration of the value at a particular string position. In the case of binary string, this simply means changing the state of a bit from 1 to 0 and vice versa In this paper we provide a uniform mutation method. This method is fust to produce a mask and select a string randomly, then complement the selected string value correspond to the position of mask where the bit value is “1”. Mutation is needed because some digits at particular position in all strings may be eliminated during the reproduction and the crossover operations. So the mutation plays the role of a safeguard in GAS. It can help GAS to avoid the possibility of mistaking a local optimum for a global optimum. 5 Results
The model given in [6] is X=Ax+Bu
where ~ 0 . 2 4 4 -0.0747 377 0
1:
Reproduction is a process to decide how many copies of individual strings should be produced in the mating pool according to their fitness value. The reproduction operation allows strings with higher fitness value to have larger number of copies, and the strings with lower fitness values have a relatively smaller number of copies or even none at all. This is an artificial version of natural selection (strings with higher fitness values will have more chances to survive).
B=
-0.1431
0
0
0.0747
0.Wl
0
0
0
0
0
0
0.046
0.13
398.58
-3967
-0.178
-0.146
0
-0.M
-0.455
0.244
0
-398.56
-19498.8
-50
0 0
0
0.178
-0.0433
0
-0.2473
0 0
0
0
0
376.99
0
0
0 0 0 0 0
0.056
0
0
-0.0565
-0.3061
0.149
0
-677.39
0
0
677.78
-133M.16
-50
ro o o
10
o o
0.1234 -10234.22
2500
o
o o 0 o o o
0
1’
25001
The eigenvalues of the power system are as follows:
-25.1741+j67.8187 - 0.0904 f j9.843 - 0.2443
4.2 Crossover
Crossover is a recombined operator for two high-fitness strings (parents) to produce two offsprings by matching
-25.23925 j30.3072 - 0.0006
To improve the system damping using the two-level scheme, the decomposed system and control matrices are
517
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
as follows: For system 1
Sliding
X , = [Am,
Aeql AVJ
-0.244 A , = [ 377 0 0 El =[0
Controller
-0,747 0 -0.046 -398.56
0 0 2500
-0.1431 0 -0.455 -19498.8
Reduced Order Model
I
A8.1
I L
0.244 -50.0
1'
Sliding Controller
Reduced Order Model
Using the expression given by [7], a reduced order model of system 1 is obtained. XI, = Fix,,
+ G,u,
(18)
where xlr = [Am, A8,
'1
GI =[-0.1826
1'
-0.734
4
=[
-0.26 377
-0.07
1,
Fig. 3 The structure of the proposed design
and
I n t h paper, we propose the genetic algorithm to fmd the switching surface vector and the switching control signals. Furthermore, in order to find the parameters to facilitate the controlled system with small integml absolute error, we define the following fitness function:
For system 2
x2=bo2 as, A,
=I
f =g1*g2
-0.2473 -0.177 377 0 0 -0.0565 0 677.78
4 =[O 0
AV~~J
kq2
2
0
-0.3061 -13364.1
0.1492
, i = 1,2, and
-50.0
0 2500'1
6=1
Using the expression given by [7], a reduced order model of system 2 is obtained.
The following parameters of GAS are considered
i2r = F2x2,+ Gzu2
1' -2.6202 1'
where xgr =[Am2 A6,
G, =[-0.2688
F2
=[
-0.18 377
-0.18
1,
Population size40 Crossover probability=0.9 Mutation probability=0.03 Chromosome length=30 Generations=2000 Range=[-250 2501
and
The global control matrix G is given by G=[
(20)
-0.146
0.118 -0.36 0.0975 -0.2974
0.02 -0.209 0.0141 -0.1465
From eqn.(ll), and eqn.(l6), we can get the optimal switching hyperplane C and the feedback gain K , of the power system.
1
For system 1
And the proposed design is shown in Fig. 3.
s, = [-3.0056 K:l =[-350.9
-1.6189b,, = C:x,, 0.12112
]
518
Authorized licensed use limited to: Tamkang University. Downloaded on March 24,2010 at 04:51:50 EDT from IEEE Xplore. Restrictions apply.
For system 2 s2 =[-1.457
Ks:2
=
-1.1052xl, =C:x,,
[- 126.66
0.079773
]
The transient responses of the angular frequencies with global control to a 5% change in the mechanical torque of system 1 and system 2 are shown in Fig.4 and Fig.5 respectively.
“.I
\ - - - --I
Fig3 (a) The torque angle response of system 1
Fig.4 (a) The angular frequency response of system 1
Fig.5 (b) The torque angle response of system 2 Fig.5 The transient responses of the torque angles 6 Summary
This paper proposes a new approach for combining genetic algorithm and sliding mode control to design the power system stabilizers. We use the genetic algorithm to fmd the switching surface vector and the switching control signals, and use the sliding mode control to fmd the control signal of the generator. By using the output feedback only, this approach reduces the implementation cost and the reliability of the power system. Comparison of the proposed method with the traditional optimal control method, the effectiveness of the proposed method in enhancing the dynamic performance stability is verified through the simulation results.
i
Fig.4 (b) The angular frequency response of system 2 Fig.4 The transient responses of the angular frequencies
References [l] Arcidiacono V, Ferrari E, and Saccomanno F.. Studies on damping electromechanical oscillation in multimachine system with longitudinal shucture, IEEE Transactions, PAS-95:450460, 1976. [2] Bauer D. L., Cory D.B., Buhr W. D., Ostroski G B., Cogswell S. S., and Swanson D. A.. Simulation of
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